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arXiv · 2112.11412

Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals

Kaisa Matomäki·Jori Merikoski
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Abstract

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Ter\"av\"ainen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

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Kaisa Matomäki, Jori Merikoski. 2021-12-21. Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals. https://arxiv.org/abs/2112.11412

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